Finite Group Databasebeta

← #10849690

Subgroups — #10849690

Order 768.

1094 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3506C211yesmaximal, characteristic
2#83341C211yesmaximal, characteristic, complemented
3#83283C211yesmaximal, characteristic, complemented
4#82379C211yesmaximal, characteristic, complemented
5#71852C214yesmaximal, complemented
6#71841C211yesmaximal, characteristic, complemented
7#71818C211yesmaximal, characteristic, complemented
8#71664C214yesmaximal, complemented
9C4×#833C211yesmaximal, characteristic, complemented
10#25694—31nomaximal, Sylow
11C3×#559—22no—
12C3×#556—21no—
13C3×#555—21no—
14C3×#553—22no—
15C3×#550—21no—
16C3×#547—21no—
17C3×#538—22no—
18C3×#535C2212yes—
19C3×#533C2211yescharacteristic
20C3×#532C2211yescharacteristic
21C3×#525—22no—
22C3×#520C2212yes—
23C2×C3×C4×D4C2211yescharacteristic
24C2×#924—22no—
25C2×#869C2211yescharacteristic
26C2×#837C2214yes—
27#5232C2214yes—
28#5221C2212yes—
29C2×#833C2212yes—
30#5080—21no—
31#5079—21no—
32#5078—21no—
33#5070—21no—
34#5064—22no—
35#5055—22no—
36#4995C2214yes—
37#4975C2212yes—
38#4949—22no—
39C4×C4⋊Dic3C2211yescharacteristic
40C4×#208C2211yescharacteristic
41C4×#208C2211yescharacteristic
42#4532C2214yes—
43#4530C2212yes—
44#3506D311yescharacteristic
45#3165—31no—
46#3141—31no—
47#2542—31no—
48#2037—34no—
49#1986—31no—
50#1980—31no—
51#1859—34no—
52C4×#100—31no—
53C2×C3×#41—22no—
54C2×C3×#41C2312yes—
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4D412yes—
57C22×Dic6—21no—
58C3×C4⋊Q8D411yescharacteristic
59C3×C4⋊Q8D411yescharacteristic
60C3×C4⋊Q8D411yescharacteristic
61C3×#127—22no—
62C3×#127D411yescharacteristic
63C3×#126—24no—
64C3×#125D412yes—
65C3×#124—24no—
66C3×#124—22no—
67C3×#124D412yes—
68C3×#123—24no—
69C3×#123—24no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—21no—
72C3×C22⋊Q8—21no—
73C3×#121—22no—
74C3×#121—24no—
75C3×#121—21no—
76C3×#121—21no—
77C3×#121—22no—
78C3×#121—22no—
79C3×#120—24no—
80C3×C4×Q8—22no—
81C3×C4×Q8—21no—
82C3×C4×Q8—21no—
83C3×C4×D4—24no—
84C3×C4×D4—22no—
85C3×C4×D4—24no—
86C3×C4×D4—21no—
87C3×C4×D4—21no—
88C3×C4×D4—21no—
89C3×C4×D4—21no—
90C3×C4×D4—21no—
91C3×C4×D4—21no—
92C3×#117—22no—
93C3×#117—22no—
94C3×#117C2312yes—
95C2×C3×C4⋊C4—22no—
96C2×C3×C4⋊C4C2311yescharacteristic
97C2×C3×C22⋊C4D412yes—
98C2×C3×C42C2311yescharacteristic
99C2×#217—24no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3C2312yes—
103#924—48no—
104C2×Dic3⋊C4—24no—
105C2×C4×Dic3C2312yes—
106C2×#208C2312yes—
107C2×#208C2312yes—
108#869—21no—
109#869—21no—
110#869—21no—
111#869—21no—
112#837—28no—
113#835—24no—
114#833—24no—
115#832—24no—
116C3⋊#95—21no—
117#808—24no—
118#807—24no—
119#803—21no—
120#803—21no—
121C4×C3⋊C8—21no—
122C4×C3⋊C8—21no—
123#559—22no—
124#556—21no—
125#555—21no—
126#553—22no—
127#550—21no—
128#547—21no—
129#538—22no—
130#535C2×D312yes—
131#533C2×D311yescharacteristic
132#532C2×D311yescharacteristic
133C2×C4⋊Q8—31no—
134#525—22no—
135C2×C22⋊Q8—62no—
136#520C2×D312yes—
137C2×C4×D4C2×D311yescharacteristic
138#501—62no—
139#500—61no—
140#498—61no—
141#489—62no—
142#487—61no—
143#486—61no—
144#428—34no—
145C2×#104—34no—
146#417—34no—
147C2×#100—32no—
148C4×M16—31no—
149C4×M16—31no—
150#399—62no—
151#390—32no—
152#384—32no—
153C4×C4⋊C4—31no—
154#343—32no—
155#339—34no—
156C24×C3C2×D412yes—
157C3×#41—24no—
158C3×#41—24no—
159C3×#41—84no—
160C2×C3×Q8—21no—
161C2×C3×Q8—21no—
162C2×C3×Q8C2×D412yes—
163C2×C3×D4—22no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—21no—
167C2×C3×D4—21no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—22no—
171C2×C3×D4—22no—
172C2×C3×D4C2×D412yes—
173C22×C3×C4—24no—
174C22×C3×C4—22no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—44no—
178C22×C3×C4C2×D411yescharacteristic
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D411yescharacteristic
181C22×C3×C4C2411yescharacteristic
182C22×Dic3—22no—
183C22×Dic3—22no—
184C2×Dic6—44no—
185C2×Dic6—41no—
186C2×Dic6—41no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4C2×D412yes—
197C3×C4⋊C4C2×D412yes—
198C3×C22⋊C4—24no—
199C3×C22⋊C4—24no—
200C3×C22⋊C4—24no—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—22no—
205C3×C22⋊C4—22no—
206C3×C42—24no—
207C3×C42—22no—
208C3×C42C2×D411yescharacteristic
209C3×C42C2×D411yescharacteristic
210C3×C42C2×D411yescharacteristic
211C3×C42C2×D411yescharacteristic
212#217—48no—
213C4⋊Dic3—24no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—41no—
217C4⋊Dic3—41no—
218Dic3⋊C4—48no—
219C4×Dic3—24no—
220#208—22no—
221#208—22no—
222#208—22no—
223#208#4114yescomplemented
224C2×C3⋊C8—22no—
225C2×C3⋊C8—22no—
226C2×#41—22no—
227C2×#41C22×D312yes—
228C22×Q8—61no—
229C22×D4C22×D311yescharacteristic
230C23×C4#6712yes—
231C2×M16—32no—
232C2×M16—32no—
233C4⋊Q8—61no—
234C4⋊Q8—61no—
235C4⋊Q8—61no—
236C4⋊Q8—61no—
237C4⋊Q8#6711yescharacteristic
238C4⋊Q8#6711yescharacteristic
239C4⋊Q8#6711yescharacteristic
240#127—22no—
241#127#6711yescharacteristic
242#126—24no—
243#125#6712yes—
244#124—24no—
245#124—22no—
246#124#6712yes—
247#123—24no—
248#123—24no—
249C22⋊Q8—22no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C22⋊Q8—128no—
253#121—22no—
254#121—24no—
255#121—21no—
256#121—21no—
257#121—22no—
258#121—22no—
259#120—24no—
260C4×Q8—22no—
261C4×Q8—21no—
262C4×Q8—21no—
263C4×D4—24no—
264C4×D4—22no—
265C4×D4—24no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4—21no—
272#117—22no—
273#117—22no—
274#117C22×D312yes—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—32no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—64no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—64no—
282C2×C22⋊C4#6712yes—
283C2×C42—32no—
284C2×C42C22×D311yescharacteristic
285#107—64no—
286#106—64no—
287#104—68no—
288#103—64no—
289#102—64no—
290#100—64no—
291#97—61no—
292#97—61no—
293C4×C8—61no—
294C4×C8—61no—
295#95—61no—
296C23×C3—22no—
297C23×C3—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—42no—
301C23×C3—42no—
302C23×C3C22×D411yescharacteristic
303C3×Q8—24no—
304C3×Q8—42no—
305C3×Q8—81no—
306C3×D4—24no—
307C3×D4—42no—
308C3×D4—42no—
309C3×D4—42no—
310C3×D4—42no—
311C3×D4—42no—
312C3×D4—81no—
313C3×D4—84no—
314C3×D4—84no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—41no—
321C2×C3×C4—44no—
322C2×C3×C4—44no—
323C2×C3×C4—42no—
324C2×C3×C4—82no—
325C2×C3×C4—84no—
326C2×C3×C4#12011yescharacteristic
327C2×C3×C4#12011yescharacteristic
328C2×C3×C4#12011yescharacteristic
329C2×C3×C4#12011yescharacteristic
330C2×C3×C4C22×D411yescharacteristic
331C2×C3×C4C22×D411yescharacteristic
332C2×C3×C4C2×#4112yes—
333C2×C3×C42−1+412yes—
334C2×Dic3—24no—
335C2×Dic3—42no—
336C2×Dic3—42no—
337C2×Dic3—44no—
338C2×Dic3—42no—
339Dic6—82no—
340Dic6—82no—
341C3⋊C8—24no—
342C3⋊C8—42no—
343C24C2×#6712yes—
344#41—24no—
345#41—24no—
346#41—84no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—124no—
350C2×Q8—121no—
351C2×Q8—121no—
352C2×Q8C2×#6712yes—
353C2×D4—22no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4C2×#6712yes—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—44no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4C2×#6711yescharacteristic
371C22×C4C2×#6712yes—
372C22×C4C2×#6711yescharacteristic
373C22×C4C23×D311yescharacteristic
374M16—34no—
375M16—62no—
376M16—62no—
377M16—62no—
378C2×C8—62no—
379C2×C8—62no—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—64no—
390C4⋊C4—62no—
391C4⋊C4—62no—
392C4⋊C4—128no—
393C4⋊C4—121no—
394C4⋊C4—121no—
395C4⋊C4C2×#6712yes—
396C4⋊C4C2×#6712yes—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—128no—
406C42—24no—
407C42—22no—
408C42—64no—
409C42C2×#6711yescharacteristic
410C42C2×#6711yescharacteristic
411C42C2×#6711yescharacteristic
412C42C2×#6711yescharacteristic
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—44no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—82no—
423C22×C3C2×#12011yescharacteristic
424C22×C3#53211yescharacteristic
425C22×C3#53611yescharacteristic
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—42no—
430C3×C4—42no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4#45312yes—
434C3×C4#54712yes—
435C3×C4#54812yes—
436Dic3—44no—
437Dic3—82no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—42no—
443C23—42no—
444C23C22×#6711yescharacteristic
445Q8—24no—
446Q8—42no—
447Q8—81no—
448Q8—242no—
449Q8—242no—
450D4—24no—
451D4—42no—
452D4—42no—
453D4—42no—
454D4—42no—
455D4—42no—
456D4—81no—
457D4—84no—
458D4—84no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—64no—
469C2×C4—82no—
470C2×C4—84no—
471C2×C4—122no—
472C2×C4—122no—
473C2×C4—124no—
474C2×C4—122no—
475C2×C4#95311yescharacteristic
476C2×C4#95311yescharacteristic
477C2×C4#95311yescharacteristic
478C2×C4#95311yescharacteristic
479C2×C4D3×#4112yes—
480C2×C4#101012yes—
481C2×C4C22×#6711yescharacteristic
482C2×C4C22×#6711yescharacteristic
483C8—64no—
484C8—122no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—82no—
488C2×C3—82no—
489C2×C3#243211yescharacteristic
490C2×C3C2×#45311yescharacteristic
491C2×C3#302211yescharacteristic
492C22—21no—
493C22—21no—
494C22—44no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—82no—
502C22#582511yescharacteristic
503C22#583211yescharacteristic
504C22C2×#95311yescharacteristic
505C4—21no—
506C4—21no—
507C4—21no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—42no—
512C4—124no—
513C4—242no—
514C4#524112yes—
515C4#582412yes—
516C4#583112yes—
517C3#2569411yescharacteristic, complemented, Sylow
518C2—21no—
519C2—21no—
520C2—82no—
521C2—82no—
522C2#8409311yescharacteristic
523C2C2×#524111yescharacteristic
524C2#8412311yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#3506C2
socleC22×C3C2×#120
hypercenter#3506D3
hypocenterC3#25694
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #83341C2semidirect
3. #83283C2semidirect
4. #82379C2semidirect
5. #71852C2semidirect
6. #71841C2semidirect
7. #71818C2semidirect
8. #71664C2semidirect
9. C4×#833C2semidirect
223. #208#41semidirect
517. C3#25694semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C3×#35062
#833412
#832832
#823792
#718522
#718412
#718182
#716642
C4×#8332
#256943